The present invention relates in general to imaging methods, and more particularly to a new method for imaging with low frequency electromagnetic fields, for interpreting the electromagnetic data using ray tomography, and for determining the earth conductivity with high accuracy and resolution.
Electrical conductivity of the earth is a very important factor in underground explorations, such as explorations for minerals, hydrocarbons and groundwater, and in analyzing environmental and crustal problems. Conventional methods approximate the electrical conductivity by using low frequency (less than 1 MHz) electromagnetic fields. The differential equation that describes the behavior of such electromagnetic fields is a diffusion equation, such that the inversion solution of the fields satisfying this equation is relatively difficult, particularly for complex models. There has been several attempts to resolve such diffusion equation.
Some of these attempts include earlier studies by G. Kunetz, "Processing and interpretation of magnetotelluric soundings", Geophysics, volume 37, pages 1005-1021 (1972); P. Weidelt, "The inverse problem of geomagnetic induction", Zeitung for Geophysiks, volume 38, pages 257-298 (1972); and S. Levy et al., "Subsurface imaging using magnetotelluric data", Geophysics, volume 53, pages 104-117 (1988). These studies nave shown a relationship between diffusion equations and wave equations, but have been limited to magnetotelluric problems in a layered earth.
Using equations for scalar fields, M. Lavrent'ev, V. G. Romanov and S. P. Shishatski, in "Ill posed problems of mathematical physics and analysis", Nauka, volume 64, pages 250-252 (1980), presented a mathematical transform. between fields satisfying the diffusion equation and its corresponding wave equation. Subsequently, G. A. Isaev, and V. V. Filatov, in "Physicomathematical principles of visualization of nonstationary electromagnetic fields", Geol. i Geofiz., volume 22, pages 89-95 (1981), and V. V. Filatov in "Construction of focusing transformation of transient electromagnetic fields", Soviet Geology and Geophysics, volume 25, No. 5, pages 87-93 (1984), tested the imaging concept, using controlled-source electromagnetic data, with limited success.
In yet another attempted solution, K. H. Lee, Gan Quan Xie, and H. F. Morrison, in an article entitled "A new approach to modeling the electromagnetic response of conductive media" Geophysics, volume 54, pages 1180-1192 (1989), generalized the transform to include vector electromagnetic fields and arbitrary sources.
Yet another study by K. H. Lee, in an article entitled "A new approach to interpreting electromagnetic-sounding data", Lawrence Berkeley Laboratory, Annual Report 1988, LBL-26362, pages 24-27, showed that the construction of the wavefield with reasonable resolution would require approximately four decades of time-domain, or frequency-domain data, with a maximum allowable noise in the power spectrum of about three percent (3%). However, useful information, such as the traveltime, would remain in the transformed wavefield, even if the time window is substantially reduced.
The foregoing attempts have met with varying degrees of success, in obtaining high resolution earth electrical conductivity distribution useful for underground explorations and environmental studies. However, no practically useful method has been proposed for high-resolution imaging of electrical conductivity using the wavefield transform technique.
Therefore, it would be highly desirable to have a new method for imaging with low frequency electromagnetic fields, via a wavefield transform and for determining the earth conductivity with high accuracy and resolution.